Abstract
A meshless method based on the singular boundary method is developed for the numerical solution of the time-dependent nonlinear sine-Gordon equation with Neumann boundary condition. In this method, by using a time discrete scheme to approximate the time derivatives, the time-dependent nonlinear problem is transformed into a sequence of time-independent linear boundary value problems. Then, the singular boundary method is used to establish the system of discrete algebraic equations. The present method is meshless, integration-free, and easy to implement. Numerical examples involving line and ring solitons are given to show the performance and efficiency of the proposed method. The numerical results are found to be in good agreement with the analytical solutions and the numerical results that exist in literature.
Highlights
The sine-Gordon equation is nonlinear and has drawn considerable attention as it comes up in a broad class of modeling situations such as nonlinear optics and solid state physics [1, 2]
The soliton recovers its straightness when t = 18. These graphical results match those obtained by the finite element method (FEM) [7], the finite difference method (FDM) [8], the boundary element method (BEM) [9, 10], the differential quadrature method (DQM) [11], and the radial basis functions (RBFs) [21]
When t = 12.6, the ring soliton appears to be again in its shrinking phase. These graphical results match those obtained by the FEM [7], the BEM [9, 10], the DQM [11], the RBF [21], and the mesh-free reproducing kernel particle Ritz method [23]
Summary
The sine-Gordon equation is nonlinear and has drawn considerable attention as it comes up in a broad class of modeling situations such as nonlinear optics and solid state physics [1, 2]. In the past thirty years, a variety of numerical methods [6,7,8,9,10,11,12,13,14,15], such as the finite element method (FEM), the finite difference method (FDM), and the boundary element method (BEM), have been used to numerical simulation of the sine-Gordon equation. Some meshless methods [21,22,23,24,25,26,27,28] have been used to obtain the numerical solutions of the sine-Gordon equation. To avoid the singularity of the fundamental solution, source points in the MFS are required to locate on a fictitious boundary outside the computational domain.
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